Skip to main content
SpringerLink
Log in

Projected subgradient algorithms on systems of equilibrium problems

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

The paper proposes two parallel and cyclic algorithms for solving systems of equilibrium problems in Hilbert spaces. The algorithms combine two methods including the diagonal subgradient method and the projection method with parallel or cyclic computations. The obtained results can be considered as improvements over several previously known methods for systems of equilibrium problems in computational steps. The algorithms have also allowed to reduce several assumptions imposed on bifunctions. The strongly convergent theorems are established under suitable conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anh, P.K., Buong, N.G., Hieu, D.V.: Parallel methods for regularizing systems of equations involving accretive operators. Appl. Anal. 93, 2136–2157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejer monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–146 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set Valued Var. Anal. 20, 229–247 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Censor, Y., Gordon, D., Gordon, R.: Component averaging: an efficient iterative parallel algorithm for large and sparse unstructured problems. Parallel Comput. 27, 777–808 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Flam, S.D., Antipin, A.S.: Equilibrium programming and proximal-like algorithms. Math. Program. 78, 29–41 (1997)

    Article  MathSciNet  Google Scholar 

  10. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  11. Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21, 478–501 (2016)

    Article  MathSciNet  Google Scholar 

  13. Hieu, D.V.: Cyclic subgradient extragradient methods for equilibrium problems. Arab. J. Math. 5, 159–175 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hieu, D.V.: Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings. J. Appl. Math. Comput. 53, 531–554 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hieu, D.V.: An extension of hybrid method without extrapolation step to equilibrium problems. J. Ind. Manag. Optim. (2016). doi:10.3934/jimo.2017015

    Google Scholar 

  17. Hieu, D.V.: Halpern subgradient extragradient method extended to equilibrium problems. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales-Serie A: Matematicas (2016). doi:10.1007/s13398-016-0328-9

  18. Iiduka, H.: Convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi-nonexpansive mappings. Math. Program. 159, 509–538 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Iusem, A.N.: On some properties of paramonotone operator. Convex Anal. 5, 269–278 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52(3), 301–316 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Konnov, I.V.: On systems of variational inequalities. Russ. Math. (Iz.Vuz.) 41, 79–88 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  23. Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal. 16, 899–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Martinet, B.: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér., Anal. Numér. 4, 154–159 (1970)

    MATH  Google Scholar 

  25. Moudafi, A.: Proximal point algorithm extended to equilibrum problem. J. Nat. Geom. 15, 91–100 (1999)

    MathSciNet  MATH  Google Scholar 

  26. Mordukhovich, B.S., Panicucci, B., Pappalardo, M., Passacantando, M.: Hybrid proximal methods for equilibrium problems. Optim. Lett. 6, 1535–1550 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Muu, L.D., Oettli, W.: Convergence of an adative penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18, 1159–1166 (1992)

    Article  MATH  Google Scholar 

  28. Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space. J. Optim. Theory Appl. 160, 809–831 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nguyen, T.P.D., Strodiot, J.J., Nguyen, V.H., Nguyen, T.T.V.: A family of extragradient methods for solving equilibrium problems. J. Ind. Manag. Optim. 11, 619–630 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    Article  MathSciNet  Google Scholar 

  31. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  32. Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Glob. Optim. 56, 373–397 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Strodiot, J.J., Vuong, P.T., Nguyen, T.T.V.: A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J. Glob. Optim. 64, 159–178 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization 64, 429–451 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. Math. Anal. Appl. 298, 279–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yen, L.H., Muu, L.D., Huyen, N.T.T.: An algorithm for a class of split feasibility problems: application to a model in electricity production. Math. Method Oper. Res. 84, 549–565 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The guidance of Profs. P. K. Anh and L. D. Muu is also gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    Dang Van Hieu

Authors
  1. Dang Van Hieu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dang Van Hieu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hieu, D.V. Projected subgradient algorithms on systems of equilibrium problems. Optim Lett 12, 551–566 (2018). https://doi.org/10.1007/s11590-017-1127-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-017-1127-8

Keywords

Search

Navigation